Combined transformations

The order

If P is applied first and then Q, the single matrix is QP: the image of v is Q(Pv) = (QP)v.

Read the product from right to left: in QP, P acts first.

Order matters

Rotation 90° anticlockwise R = (0 -1; 1 0) and reflection in the x-axis X = (1 0; 0 -1):

• X first, then R: RX = (0 1; 1 0), reflection in y = x.
• R first, then X: XR = (0 -1; -1 0), reflection in y = -x.

Two reflections combine to give a rotation, and a rotation followed by a reflection gives a reflection.

Repeated transformations

Applying R twice is R². Two rotations about O through θ and φ combine to a rotation through θ + φ, so for example (rotation 45°)² = rotation 90°, and (rotation 90°)⁴ = I.

Worked example

Find the single matrix for: reflection in the x-axis followed by rotation 90° anticlockwise about O. Describe it.

1. X = (1 0; 0 -1) first, then R = (0 -1; 1 0).
2. The combined matrix is RX.
3. RX = (0 1; 1 0)

Answer: (0 1; 1 0): reflection in the line y = x

Check yourself

1. Transformation P is followed by transformation Q. Which matrix represents the combined transformation?
   • PQ
   • P + Q
   • QP
   • P^-1Q
   Show answer

   QP. The image of v is Q(Pv) = (QP)v, so P must be next to v.

2. R = (0 -1; 1 0) and X = (1 0; 0 -1). Find the top-right entry of the matrix for 'R followed by X'.
   Show answer

   -1. R first, so the matrix is XR = (0 -1; -1 0).

3. Describe the single transformation equivalent to rotation 90° anticlockwise about O followed by reflection in the x-axis.
   • Reflection in y = x
   • Rotation 90° clockwise about O
   • Reflection in y = -x
   • Reflection in the y-axis
   Show answer

   Reflection in y = -x. XR = (0 -1; -1 0), which sends (1, 0) to (0, -1) and (0, 1) to (-1, 0).

4. A stretch parallel to the x-axis with scale factor 2 is followed by reflection in y = x. Find the y-coordinate of the image of (3, 1).
   Show answer

   6. Stretch: (3, 1) goes to (6, 1). Reflect in y = x: (1, 6). The matrix is (0 1; 2 0). Give the y-coordinate: 6.

5. R represents rotation 45° anticlockwise about O. What does R² represent?
   • Rotation 45° anticlockwise about O
   • Enlargement scale factor 2
   • Reflection in y = x
   • Rotation 90° anticlockwise about O
   Show answer

   Rotation 90° anticlockwise about O. Two rotations of 45° about O make 90°. Check: R² = (0 -1; 1 0).

6. R represents rotation 90° anticlockwise about O. What is the smallest positive integer n with Rⁿ = I?
   Show answer

   4. Four quarter turns make a full turn.

7. A reflection is followed by another reflection in a different line through O. What kind of single transformation results?
   • A reflection
   • A stretch
   • An enlargement with negative scale factor only
   • A rotation about O
   Show answer

   A rotation about O. Each reflection has determinant -1, so the product has determinant 1, and it preserves lengths: it is a rotation about O.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics, Further Mathematics and Pure Mathematics specification (issue 3), Unit FP1
• Pearson Edexcel IAL Further Pure Mathematics 1 scheme of work

Last updated 8 November 2026. Spotted a mistake? Tell us.